:-..

ARB No. 14C16

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ': I '

ORIGINALLY ISSUED

October 1944 as Advance Restricted Report L4016

COMPRESSIBILITY EFFECTS ON HEAT TRANSFER AND

PRESSURE DROP IN SMOOTH CYLINDRICAL TUBES

By Jack N. Nielsen

Langley Memorial Aeronautical Laborator,y Langl~y Field, Va.

WASHINGTON

, /

NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid of advance research results to an authorized group requiring them for the war effort. They were pre­viously held under a security status but are now unclassified. Some of these reports were not tech­nically edited. All have been reproduced without change in order to expedite general distribution.

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https://ntrs.nasa.gov/search.jsp?R=19930093523 2020-03-03T03:22:04+00:00Z

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!;TACA AHR Fo. I4C16

Nt.TIONbL ADVISORY COMTHTTEE FOR AER.ONAUTICS

ADVANCE EF.STHICTED REPORT

cmr:PRESSIBIIJrry EFFEC'rS O:\f HEAT TRANSFER AND

PRESSURE DROP IN SMOOTH CYLINDRICAL '1'TJBES

By j-ack N. Nielsen

SUMT.1AHY

An analysis is made to simp11fy pressure-drop calcu­lations for ncnadiabatic and ad1.ahatic friction flow of air in smooth cylindrical tubes wl-~en the densIty chanses due to beat transfer and DreGS~~e dro0 are G?Dreciable. Solutions of the equation"of motton a~e obtai~ed by the use of ~eynolds' analoGY between heat transfer and skin friction. ~hart~ of the solutIons are presented for making pressure-drop calculatio~s. A technique of using the charts to determine the poe·.t tion of D normal shoc 1.{

in a tube io described.

INTRODUCTION'

The heat transfer and pressure drop for the flow of air in smooth cylindrical tubes may be calculated with­out difficulty only when the density changes that ac­company the changes of temperature and pressure are relatively smell. In the present paper en attempt is made to analyze flow witb large density cbanges, such as occur at high 'Iach numbers, in order to simp15.fy the calculation of such flow and to provide a basis for the correlation of data.

The analysis deocnds on the validity of Reynolds' analogy betV'!een heat tr&nsfer ::md sl\1n friction, which leads to the equality of the heat-trenofer andsktn­frlc tion coefficients. by n:c: ,°1.113 of thl s sir;~plif:i.cation the equation of motion car:. be solvod for the nonadlabatic case and th98tagnRtlo~-tem~?ra~~re variation along the tube can th311 be 0 Ltc -; led as a fu:lC ti on of the Mach number; the solution is f1ven in the form of a chart. The adiabatic case, a limiting form of the nonadiabatic case, is also· analyzed in 30~e detail and the solution

2

for the pressure variation is given in grap~ical form. Some discussion is j ncluded of available ex:?orimental data (references 1 to 3) in order to ev~luate the as­sumptions that are made hereln and the applicability of the solutions.

. SYMBOLS

T ab~olute temperature, OF + 1~.60

x distance in flow dj.rect1on, feet

y dlRtance pernendlcular to f100 di~ection, feet

u velocity in flow directicD, fret per second

v velocity perpondtnular to flow direction, feot per secone

k thonr.al condnctivit~.'", Btu ---------------

soconds·x fcct2 x °F/foet

p mass density, clu~3 pe~ nubic foot

b 1 t j • t ___ 8.1 ufS_E._, __ a so u e v·.scos: ~J -

se(~(;nds Y feet

Pr Prandtl nurrber (ao')ut J.T; for laminar .:Clow of air at room temperature)' (tJ.cp/ld

Ts st~lgr,at ion ternperatul'e, or +' J.!.60

C nondirrensional heat-transfer coefficient H

OF nondjrrensional ~kin-friction coefffcie~t

D tube diameter, ie0t

G mass rate of flow ryor unit ar3B, slu£s ___ _

fecd 2 x seconds

Re Reynolds nUlr.oer

p static prossure, ?ounds per squtre foot

t

NACA ARR No. L4016

R gas t t f00t-pounds cons an , . -slugs x of

also tube radius, feet

3

(about 1716 for air);

J mechanical equivalent o'f heat (778 foot-pounds per Btu)

specific heat at constant pressure, Btu

slugs x of

M

r

A

K

ratio of s0ecific heats (taken as 1.404 for all figures) ,

l.Iach number (hRT) wall te!!1p erature, of + h60

nondimensional terrperature ratio (Ts/Tw)

distance from tube axis, feet

tube cross-sectional area, square feet

tube length, feet

isentropic mass flow per unit area in throat of supersonic nozzle, __ 'ylll:g~..;..s __ _

feet2 x seconds

constant

Subscri-pts:

c tube center

o initial

1 before shock

2 after shock

s stagnation

A single bar over a symbol denotes an average with respeot to cross-sectional area~ a double bar denotes an average with respect to cross-sectional area with pu as weighting factor.

4. T,TA.""" AT:''= ":IT" Il,C]O~ J. ... vH '.\~, .,',1 • . .t.t .

RE:':KOLDS' ANAT.JOGY

Reynolds' analogy between hent transfer and skin fr::1.ct:i.on involve~1 a s':'milari.ty ~f velocity and tempcr 13.­

ture fields. For stead~ 10w-sp0ed flow of ~ir a10ng d

flat LJlate, the voloel t-y and temperature fie] ds d.re gj ven by the simolj fied boundary-1Ci.yer c,1.uations

'1 aT aT 11 oc..T u-- + v-- - --ox aJ ~;Cp d,y2.

.1

and

au au 2 l!.:. Li! ( ]. ) u·-_· -I- .,- = \ ,

QX o:·.,r P ? oy-

If the Prandtl n'..lmbcr ;:,c p/.<: is "..1ni t:r ar.d if the plcte i8' at uniform te::J.r.' cx·atUi'0 TvI" the ('r - TV[) -field and the u-field are 3iwl1ar, sin~e their dlfferenti81 equations a~1r1 bOund8!'y concU J.:;j O:1S et'6 thon tdentlcal. piaI' lamlner flow of al~ et 0:['(11.11.3.11

:; tem;:Jcratures trIe value of· the Prandtl n'.unber is approxirr.o.te:>ly 0.75, For turbulent flov', h01J'Jever, the cff.;ct:tve vAlues of i-L and k are inc1"eBoed bocausa of th6 n8t tllrbulent interchange .of momontmfJ ar:d 1:W(,t, 1'csryect1.vel:y. The Prendtl numoer for tur::>ule:1t flo,,",l will iTf..(ry ~.ome'l.lhut with pOAition beeaucB, on tho Lasis 0f ~i7tnG-len~th theory, t~e increase in the effect:ve va].~eG of t~G therrr:al conductivit-y Dnct tlJe visCOG"i.ty depend on the turbulent m:txing longtil €md tLo volool ty Gradj onto

.7!.ee.8\,,1re;n6nts gl ven In 1"e fer8~:c.8 ).1 Sh0W the t t.he temperature and velool t:y r.;roi'j les tn the turbulent bouncar7'l layer sl:::mg a flat nlete at unlfo1"m tem.:;orfl.­ture a~e nearJy the same even tho~~h tho boun~~ry condi tior:s of' the temperat11re End vclo0:lty fiolds were not exactly ).dentical (i'.lrin;~£ tr)G mr:l8Sl.l.reJ)1ent. T:19 measUrem6nts Indicate that the US0 of an avorage Prandtl number oj' uni. ty for 8. turbult.mt bO'lnciary 1::.tJ8I· sbou1d not be greatly in error. The rr~anuromant8 in refer­ence 4 a1.so show that. the e,fj"'ec.t of' Gsns:i.ty C;:r:::-.d:i.8nts on the veloci ty profile .mo.~r be ~1ee:lc')tr)d ae moderately small temperature differer:ees.

NACA ARB No. r~c16

Reynolos! analogy Illay be dlrectly extended to 10\1''/­

speed turbulent flow of air in tubes if the eff'ecto on the velocity profIle of the pressure gradients due to skin i'ri.ction and of the dcnsi ty [;radients due to beat trf,nsfer ID8::;r be eli sregUl·d~d. This aSSl.urpt:1.on is

5

jUf,tiLLed by the good expel'lmento.l agroement of tbe lo~-speed skin-friction and heat-transfor data in refer­ence 5 with neynnl~s' nnDlogy.

The effect of com:Jress.ibil'lty on the ve10city pro­file has been measursd by Fr~ssel (referenco 2), who concludes that for' subsoni.c 8.diubs.tic flow of aLl" in smooth cyl:1.ndrical tuoes tho velocity oroflle is not affected by compress:tb:Dity. l1'h8 effect of COTI1T}:ees-s: blli tJ' on the te:n:per'zl ture ':)I'ofil e may be seen from the following differential 8c:unt1011 for two-dImensional motion of a comryY'e8sible fluid ln a boundary laY8r:

Equation (2) is 8iven by Goldstein in referenc~ 6 and attribtlte,:1 by him to Busemo.nn (referen~e 7). for PI' = 1 and for stGady flow,

oTs oT~ k 021":1 13 (3) u-- + v-'" = -----? ox a"s pc oY'-p

C omparl s on of eql1ati ons (l) fl.nQ (3) shows thtl.t f')r c On1-

Pl'8 S 8i ble flow the (T~1 - ']1',"1) -f1. elel and the u-flel dare .' similar for identical boundury C0ndltlons. For compres­sible flow Rcynolrl.s' analogy thu3 r:>ostulat'38 the 8irr:t­lari ty of the stagnatlon·-tl3·.n)crat1.1.l'e and. voloei ty fi91ds. F'or in:Jorr.nressiblo flo'N this wlmllnri t~v' red1.l.CGs to t.hs: similEJ.I'ity ,)f the (rr·- 'I'w)-fjeld and the u-field.

Reynolds' enology rc~~lts in ~n equnl!ty of the n ')noimenpional heut-tr::msfer o.nd sldn-i'::,'lctlon c0'Ji.'fi­cients, v!hi8h is the bt::\sis f·;l' th0 f.ubsoquent analysis. The nond-lmensional ~o'3ff:iclellts aro c-ivol1 b:r the followinE, e;·~~)res sj_ ons!

6 rACA ArR No. rl~C16

k[O(TW :..!~)J oy . - - y=~ ( L.)

puc", (~". - ':2 <' ) ""' vv ..,

Th8 single bar denotes ~lJ.1 aV'3ro.ge 'Nith respect to oross­secti onal aro!::t. The h0 0. t _. tI·ansfer C oeffioient is the ratio of the heat transfer per unit aroa per unit tlme at the wall to the average total energy (relative to the energ-y wJ:ere Ts == Tw) of the :'luid. flowinE, through unit crocs-sectional aroa PCI' unit tirre. The okin­friction coefficient 13 the ratio of tho sr~3lJ.ring stress at tbe \':0.11 to trlG ~~v{)r::tge mOTl'.enturr' of the fluid floVv'ing thY',O'.lGh uni t cr~SS-83C LionCtl 8re8. ')81' unit tirl'le. '1'he sirr:ilar-tty of tLe (Tv.' - rrs)-fleld and the u-field gives the follo~in8 relationship:

( 6)

From equations U+) to (6),

'An experimental verification of this relationship for in00npressible flow ie gi ven in reference :5 from the data of several inveotigutors.

The value of eM' ~.'Jill var:f with Reynclds number, roughness of tube wall, :.~nch r:~.lr~bE)r. sbape of tub8 entrance, e.nd distance fro!1~ tube entrance. For odiabetic flow in smooth tubes Leyond thA entrance lercgth in which the velocity profile 48 still chonging, Fr~ssal In ref­erence 2 nhows that Gy;' io in ler;endent of r,":8.ch numrJor

NACA ARR No, 14c16 '7 I

'" '" up to the speed of sound and is eiven by the von Xarman-Nikuradse relationship

1

/8CF

= -0.8 + 2 log Re y8c.p (7)

In computing the Reynolds number for the flow in a tube, the average air temperature may be used for evaluating the viscosity.

Pressure-drop and hent-loss measurements for hot air flowing through a water-cooled tube are reported in reference 1. For subsonic flow, it was found that

" Reynolds' analogy is independent of !Lach number if the heat-transfer coefficients are calculated on the basis of the stagnation temperature and that the values of the skin-friction coefficient'are in accord with formula (7).

Pressure-drop and heat-loss measurements for burned oil-air mixtures flowing through a smooth water-cooled tube at hiRh temoeratures and velocities are given in reference o. The measurements are in good ac60rd with Reynolds' analogy. It is also shown that, under the test conditions of reference 0, the skin-friction coef­ficients are gi van by the von Yarman-:[Jikuradse relation­ship for incompressible flow (equation (7)).

NONADIABATIC FRICTION FLOW

The equality of the heat-transfer and skin-friction coefficients given by Reynolds I analogy is used to reduce the differential equation of motion for compressible flow in a smooth tube at unlform wall temperature from three variables to two variables. The resulting equation is solved by the isocline technique to give a chart for making pressure-drop calculati0ns. Particular attention is given to averaging the variables in the following derivation and solution of the equation of motion:

The differential equation of motion for uniform pressure at every cross section is

d(P + ;"2) + ~C;;,'~u2 dx = 0

8 rACA A2R 1'10. ul.c16

--The variable pu2 1 s eliminated by. introduc::Lng the average squared 'Mach number

whlch glves

= pu2

'rP

[U2

dA 'YP

'2) L ,2 d (n + Ni',.1,r + '0 IV'"')T\ dx = 0 '" ItJJ. IS IF 1"--'

or, in different form,

d. log p2 + d. log' n2 + he dx = 0 D F

(8 )

The term is r~placed with a nondimensional tem-

perature ratio by equating the differential hoat transfer from the ~all to the differential heat gained b~ the fluid; that is,

C pu( rr H w

NACA ARR No. L4c16 9

~C dx = d PlJ.Ts D H ---'-'

pu( Tw - Ts)

d~ = -_.-1 - ~

= hCF c1.x

D (10)

The double bar denotes an average with respect to cross­sectional Bren wi th ru as 'fJeJ.ght:Lng factor. Introducj.ng the last equality of eqaation (10) in equation (9) gives

~ (1 + 1\ d log p2 + d log fiTe - d lO[J ('f - 1) = 0 "1l:2 )

(11)

The last step in the derivntlon is to replace log p2

wi th an expre s'sion in '{ and i'F2. Wi th the aid of the relationshlps

P = PRT E: Ts

= Tw

')

~.12 ut.:.

= ~RT

The transformation to an expr'esslon in € and 1.12 depends on the considerations that the density and

(12)

10 rAC!~ ARR l~o. Ll.!.C:!.6

velocity profiles are given by the f- and I\12_profiles, since

T =

and

€ TIJII------

1 + _~L:._.l M2 2

These cons1.derations, in conj~'::...~ction ~J'.T1.th equatlons (D)

end (10), show that and de~end only on the €-

end M2-9rofiles, so that equatIon (12) may be written

') pc... = (13 )

where tbe cons tant K, c 10 se to 1..1.1:.i ty, a e;::~1d S onls on the E:_ and :,~2-prof:t1es, or on € and u2.

. . The constant K 11[18 becn evalll:1ted fo:!' tta or~8-

seventh power v810city nrof11e, ~11ch ie charactoristic of fully devolo~ed tu~bulant flow, given by

( _",1/7 J. • u =:: u,.. 1 - --)

v \ :\t

NACA .ARB No. liLC16 11

and for the corresponding sim1 1ar stagnation-temperature pr~file, givcn by

~ r\1/7 1 - ( = (1 - ~c)\l - R)

The evaluation of K introduces no particular mathe­roatical difficulties but is tedious. The values are

~,

plot1ed in figure 1 aga:tnst He. for severEd values of f,!.. Prom figure 1 the variation of K with M2' and ~ is seen to be second order and will be neglected. The vari.ation of !{ from t;;he values of figure 1 for' a full;; develo,ed veloci ty p;rofl1e to a value of unj.ty for a plane velocity profile is small; therefore K will n,ereafter be ta1{en constant at unity, Taking Y con­stant is equivalent to assuming one-dimensional flo~,

"2 where £" e.nd M are significant avernges.

1r;ith the simolificaLt.on that Y is constant, equation (13) may be written

log p2 = log K G:RTw + log' - 10g~'12 ~ + '( ;.-1. !;!'9] or, in differential form,

dlog p2 =dlog '1' -dlOglM2.~ + 'Y; 1 ~~ (14)

Finally, introduoing equation (lh) in eq1'l.atlon (11) gives the desired differential equation of, motion in

E an a 1'12 , Vl hi chi s

'Y -2

12 FACA ARIZ Eo. L~_C16

Equation (15) m~y be rcgardGd as a differential form of tho') equation of motion W:leD the C011di tions of Reynolds I analog~r ar8 fulfi.llE-d. Solutions 'were;; obtained by the ~socline technique becnuse 9 close~ solution was not found. T:'1e resultlng ct'.rvos a.rc ~hoi"m in figure 2. The lower family .)i' curves, for Of. < 1" corl'eoponds tQ heating of t.he fluid; the up~)er fUr-lily of curves, for _1 > 1, oorresponds to cooling of the fluid. If ~ > 1, € deereaser. along the tu"ue; 8 1.1d, tf ~ < 1, = E increases along the tube. F~r s~bsonjc entrance ~~e.ch numbers, the local .r.:ach num~Jer' 111~reasec wi tb tube length to the li:ni tine; vD.lue of uni ty nne, for su:~ersonie entrance ~ .. ~ach numbE'rs, the 10csJ. ro.c:1. nurrber decreDs2 S

with tu.be length to tbe liln.1.tin:~: valuo of unity. In order that a solution :lass cont},yuolls1y t:1.roug;:1 a IV~8.ch

nur.:ber cf un! ty, t:he vaJuG of -f: w0uld hilVfl to :lnCr08.:::e for the IJx)per set ()f curve,s, 1'0'(> \':hitJ:1 the fluid Is a1re~dy hotter than the wall, ard ~ould hav~ to dccro~se for the 101Ner se t of ~urve 3, fo::' I":hich the f1 u.id is alread'J colder thr..n tho wal:.. Tnt..s'!;uch as EJUcb changes are contrary to the sor~ond J.IYV': o~ tberl1~(')dY~1amics, ttG local ftc~ch number J"'H3Y' r.:.ot ry[)SS continu.ousl:,' thrO'l)_Ch. unity.

Tn figure 2 flome e.):pe!'imental dat8 from refer-ence ]. nre included for com~H;Lris(")n \'\'1 tb the theorcticf .. l curves. T~1.e tube-·wall tenmere.ture was not uni forn' for' the experiment; therefore the data h~ve been plotted for two separate intrH'vals of tube 1ene.th vri th tLe tllbe-' wall tOl'J1pernture over e9ch :Lntervf.'.l consta;1t Elt its average value. t1·']}e a.greem8nt bet'ween the thc0ret.icCl.l and 8xpcrimental results 1.8 C1080 .. 3nlOoth curve;:::: instead of eJ-:'pe:::>:imental points a:["e ~l.:i vEtr. 121 figure 2 because tho experirrentul ;?oints In fl,}u'cs :3 and h of reference 1 are tJonnectcd ~y smooth curves.

In order to l'J1ake nora Gccurnte prcsBuro-dr2? calculations for heat exchangers, diacrDIDs of r plotted

/-:-) 1/2 against . \JE~ €)nlRrced from :{'i.gure 2 3.1'0 gi von in figure 3(a) for subs:,m .. tc l'3.oat-ln;:; ['.1'\ e1. 111 fll~"i.l.re 3(b) for sul)s~)nic 00011n:3' ThG CUT"TP,S of f.i.[,J)'c 3 'r:er'J ')bt.air~ed b-;y ta~{ir!3 the C':lr'ITe s of fi/-;ure 2 G.S r trc', t np[Jro:dJ'l':f.lttc,n[l. Spe'Tie.1 accurncy W8S l:lt.tr.:lno(l l.n i':he CUT'ves by- :ldJust:l.ng t~em so tbat the lnteGrDl of ths slope ~lonc each .curvo between any two ryoints

NACA ARR No. rll-C16

as given by equation (15) " equaled the difference in ordinates between the two p~lnts un the curve.

B:,r equn tl on (13), VIi th constant,

( 16)

In order to simplifJ prosE'~~r~-dro9 caloulations, lines of constant

13

have b'3en plotted :in flc;ure 3( C\.) and des1(;natcd lines of constant relative pressure. Tn making a oressure-frop calculation, the value of this ex)rcssion for the final condition is divided by the value for the initial oon­dttjon to [ive p/J(l' A r~sl1m6 I)f the ~)rOCedUl'e fol..lows.

The value of (0 for the ini t'j a1 point is found from the value S 0 f the tuLe -wall totnEerature Tw ann. the initial stagnatIon tC;JJ1Oerature f['s' which eithor

. , '- 1/2 E'T'G known. or may be !)'6ssured. Ths value of (r!102) for the lnittal point is celculat8d frJm equation (13) for vBlu08 of G, Tw' and :)0' wh~ CD 0i. ther are known or mEly b(~ measur6d. rr'h:; .values of (I) and

1."'.02)1/2 locate \~. the initial point in figure 3. The final point is located by followIng the curve of the solution

- -_. 1/2 _ through point ~o' (1IIo2) to the final value of f,

This value of f is obtained fr:)ln an integration of equa­tion (10). by assuining C

H cor-stant at its average value;

thus,

(17)

The values of CR for fully developed turbulent flow in

smooth tubes, as given b~ equation (7), are plotted in fj.eure 4., For the entrnLce lengtl1 of a tube in which the velocity profile is changing, the 'value of Cu varies

.,

along the tube nnd. depends on the entrance concii tiona. which determine the initial velocit~ profile. If experi­mental values of eR for spocific entrance conditions are not available, the va]ue of Cu given by fIgure L

.tJ

may be used as the average value for tile tube with the interpretation that this value will be slightly low. Finally, for the initial and final points now located in figure 3 the initi~l and final values of

are deterrrtned with the help of the lines of constnnt relative nrcssurs, and the uressurfl rHt~o p/~o is calculate~ by equation (16):

As an exnmple, suppose the following values are known:

_f.

NACA A'?R No. L4016

= 105 E: = 0.50 Be :!: 0

G = 1.0 L = 60 D

Tw = 7600

F abs. Po .- 2070 Ib/sq ft

Equation (13) gives

and

If the average value of Cu f')r' the tube j s ta:.um. f-\3

0.00225 from fi3ure h, theUvalue of' € computed from equation (17) is

\ - lJ. ( 0 • 0022 5)( 60 ) lIe .

Following the curv.e. in figure 3 throlJ.gh the point 0.325, 0.500 to ~ = 0.709 lo~ates tho point correopondlng to tube ex1.t.

The value s of and

15

as given by the lines of constant

16

relative pressure are 2.1)~ and 1.66, respel~tively; therefore

2_ = 1. 66 'Po 2.lh

= 0·775

The static pressure at tho Gnd of tbe tube i8 77.5 per­cent of the static pr~s~urA fit the t~lbe e~trunce.

As prl3v1.ously mentioned, thl3 8vel~D.ge v8.1ue of Crr • .1.

for the tube is nl1.ghtl:; higher tLa21 the value given by firrure 1..1.' because of t:16 lmde"v'eloned flow in the

"-.. ' L

entrance length; therefore, slig"!.l.tly lar8.~el'" valuef.: of t:

Rnd of pressure drop th~n tho foreaoing may be antici­pated. Curves for constant values of

1-:-;\1/2 are plotted 1.n figure 5 ever t"h9 l~[lnge of \I.~,-) for whi.ch nn accurate deternir:ation 1s r.ot possible from f18:,ure 3.

calClllation3 may be made for t"l,lb(~s of nonuniform w&..11 tomperature by p01"for'.n1rlg the calculat"ion foy' DeveraJ. intervals of the length on tho assumption that the tllbc-wall temperature is uniform at i ts aV6r.a[~e value over euch intervaJ. The validity of the ~rec8ding analysis for M > 1 n~y n~t be determined UJltil B7peri­mental data on Roynolds' analogy ore av&ilable for supersonic flow.

ThA preceding analysis for nonadlabatic friction flow includes adi8.b~1.tic friction flow as a sp8ci8.1 case. The equntion of. motion i or the [;J:118.bab.c case is

17

obtained from equation (15) by letting E = 1. The as-

8umpti on thu t ~ = 1 1 s based on the ex'tJGrimen tal observation that for adiabatic flow of air the tube wall i.s uniformly at stagnatio:1 temperature. F'u.cther, equation (3) for the stflgnatlon-temperatuY'e 1'ie1<4. is in accord with this o~serv~tion~ since the particular solution

T = Constant s

corresponds to no heat transfer.

The reduction of the equ&~i~n of motion (equa­tion (15)) to the ediabatic case Pl'OC'3Sc1S as follows:

From equation (10),

("'t - 1)

L~ -C,.~ dx

.1:' D

df ----~- .. - -

1,<V}:ten this Gxpre.§.sic)l1 ie i.nt:::::)r3.u~8d in equc.tion (15), together with f = 1, the r8Rultlng equation is

-ho d~ 'F 1)

Direct integration gives

The integral j_s retained in thG 801utlon because 0", J.

may not always be assQJ11ecl unlf:)rm alons the tube. The variation of OF will Dubsequently be discussed.

18 Nl:.0f. J.:.rm No. L4.ol6

Equation (18), the solution of the equation of ~otion for adiabatic friction flow, 1s shown In fig-

')

ures 6( a) and 6( b) for two subsonlc range8 cf l.:o'- and in fi£ure 6(c) for a 3upersonlc l'ange 'of j'a0

2 , liS

figures 6(b) and 6(c) show, a ,articular solutton of equation (Ie) corresponds to a 8upersonic value of

Nio 2. as well as to a subsoni c vlllue of 11102 ', These

ftgures differ only in the lines of constant pressure re.tio, wh"!.ch will be discussed.

lJ'he solutions are quali tat3.v81y liko th~~.~ ... ..r0r

~ < 1 in figure 2. For subsonic values of Mo2 the local Mach number in0reases with tuba length to a li.m.it::'ng value of unity, 'and fe)l'" supersonic vallles

of l'I02 the local :i;Tach nurnber decreases wLt'ri tui)€)

length to the limitinevalue of unit:r. A real rlow

111.ay not ?ass contin~ou,OlY through ':? = 1 bec9.u.se

~ jXo dx is a maxin,um foX' th.ts cOlldltion. D F

:) ,

Defore figure 6 may be usad, th0 variation of O~ .:

with Reynolds number, rou(~hness of tube lNall, 1v18ch nun1'oer, shapE; of tube entl'ance, end 0.1. stance from tube entrance should be known. Frossol's data (ref­erence 2) 8how that, for the range of kach l':uml')cr Ie s s tr.an tmit·:r, the velocit'jl pI'ofi..lG n;a."y not be fully developed until LID = 36; and I(Gena.n and Heurr::::mn (referer:.ce 3) found th 8 t C'q dec reased alone: the tuoe until tiD was nbout 36. b::yond th" s entr8);J.ce lenGtb, CF may be constdered constant. In figure 7 experi­

mental data, ta1~en frl)m th0 resu.lts of' r~feren~e 3 fr)I' flow beTon~ the entrance lenEth, ere corr,ared with the

~ "("'

theoretical solution of e rpl1tion (LJ), 8", being tS.lccn constant at the value given by equa.tion (7). The theo­retical sol1.l.tion is In good. agroGI":ent wi th the experi­mental data.

, )

N:'OL LRR No. 14016 19

The conclusions of referCnCBG 2 and 3 for the eff~ct of cOInnrossi'!-Jility onCB, at. Su)o1'80nic spooc.s arc not in agrsernent. No effect· of r!:tlch nl-'.111bc.."'!, on Op was found in reference 2 fat' tUb8S of IJ/D ~ 25! Referonce 3 found

a decreaoe in OF wi CD ~.~ach lrLl1nber fat' tubes of L/D f h6 on the basis of data takcn bGy~nd an effective nozzle­pipe tiD of 8.bout 35 bocause of S8verc pro3sure fluctua­tions and pos8ible var~ation of velocity prof1le in the entrance length. Further ex~erim3nt may be necessary to iso1ete the compr0ss1bi.liL:r effect and en1:irl1nce effect on OF for supersonic flow.

Cur'/es of conotGnt preS8ure rCitio p/P n have been plotted in f~.gu.re 6. The Dl'eSSUre ratios have been calcu.lated from equation (16), reduced to

(

r ',2 0' \

po) = y

- 1 2 l + :1_--1,1-2 a

-II( -. _1 .) ]. +...1- M'-

2:

1n figures 6( a) ~L1d 6( b) the lines of cOllsta:1t pressure ratio are based on the suboonia value of

Ito 2 and are valid only to 1\12 = 1 becausc the flow :may not becor~e sw)sroonic for suosrmic entrance l\:ach numoEJrs. In flgure 6(c) tr.e 1ines of constant; press\.~re rG.tio are

---ry based on the supersonic value of Mo~ and are valld for

the ent.lre ran&;e of I'[ach number. The curves of figure 6(c) a:-e also valid in the'case of normal shoe l ,; becau8 e the as sumptl ons of equ~l ti on (19) do not procl ud8 shocv-.

As a nu:ner>:i cal exam-:)lc to lllustrs.te prasf.l1.1.re-drop

calculations, constder ,M02 = 0.2, Re = 10 5, and LID = 6o. Prom figure Ll-, Ct;' 7'= 0.00225. Ta~cing Op

constant for the tube as in the example for nonadiabEltic . flow gives

20

~ l L

cp dx =

= 4 x 60 x 0.00225

From figure 6(a) ~ piPe = 0.735; thus the static pressure at the end of tho tube js 73.5 percen.t of the static pressure at the entrance of the tube.

COI'flrFmSSI O~~ SEOCYS

For supers0l1ic entrance :.'Iach number's, snaIl lengths of rapidly rj sing pressure may occur 'wl thin the tube if the static pressure in tbe j"·eservoi.r bel'!ind the tvbe fells within a ceT,tain range. T'}lC pressure disturtfH1ce cnn 'be formed by the superposi ti.on of a number of pres­sure dlsturbances oriGinating in the dovmstrea:::n flow and traveling uostrearo to a ~oint where their velocity of propagation is equ~l to the stream velocity. At this point the disturbances would com~ to rest and fOI~ a compression shock. Tho 00s1 tion of the shoc~-: 1.n the ttl.be wl11 vel'y wltb the static DreS8ure in the 1"6serv8ir. Figure 6 may be used to determine the posit1onofa norn\al shock wi th the hol'O of the follo"vvi:"w; thcoretical relatlonshi1,J between thtl T.1acL. nurtJ.'bers befor.;) and after the shock:

= 2 +- :\2("( - 1)

2 Y.'~1 2 - (~. - 1.)

(2C)

The bars are omi tted ,from the foregoing equation lJecause the shape of the velooity profile in f~ont of and behind a pressure dinturbanco is not k~own and uveraces could not be evaluated.

EquatIon (20) has been nlottod in figure 6. Several experlmental pointG from Frof.'sel' s resnl ts SllOW good acc ord with the thccretl CG.I curve. T~1.e I,Iach num'Jers for the experimontal poj.nts have bc(-)":1 COjl~)UtGd from Frosse1' s

21

date, plotted In t.erms of G/G t and pips' the following relationship:

by me'ans of

(21)

where Gt is th8 isentropic mass flow ~er unit area in the throat of tho su?ersonic nozzle in front of the pipe. The nondimensiona1 equation (21) was obtained by intro­ducing the follow~.n3 expressj on for Gt in equation (13):

"\+1 ""'---

/ 2 \ "{-1 = ~p", P",I----) ,;:> .) \" + 1

,,~ .-I

The subscript s refers to the Initial stagnatlon con­ditions of the flow. FrBssel's pressure measurements were n0t'made suff:! clent-.ly close torether to determine the precise shape of the pressure distribution at the shock. For purposes of determ.ining the points of fiG­ure 8, a sharp discontInuity was assumed to exist.

An exam?le to show the method for deterMining graphically the posi tion of the nor!J1al shoc l \: is shown in figure 6(c). 'T.f a shock forms at any point in the supersonic region, the end point ()f the shock may be determined oy movinc hOY'izontally to tbe subsonic h%.ch namber gj van by figure O. The locu~, of' such end poirlts

corresponding to a gi von value of 1:[0.2 form u shock line, wh:i. ch is shown a3 a br'oken line i:1 fig~lr6 6( c) , From the end pain t of the shocl{, the f101)l7 foll QWS trw

1 TiD

subsonic sol u ti on to t.ho value of 4. OF d(~) for () ,.

the tube exlt~ the oxit static pres8ure is then given by tho linos of constant }ressuro rat·i.o. The ~aths of the solution for two Dositions of the shock al~ shown in figure 6(c). It ~ay be seen that, cs the exit pressure ·i s lowered, the shoc 1{ move S (:owns tream. f. .. ftor the exl t

I~

/

22 NACA AIm No. IJ~C 16

pressure is lowered be] ow a. part:!.cular value 1 the sb~cl\. will pass out of the tnbe and pressure equallzatlon 'Nill occur outside the tu~)e.

Because the algebraic solutj.ons of the equation of motion for adiab&tic flow are gIven, it is clear that the splutions of normal shocks for such fl~w may elsa be determj ned algebrc.ically. Com;:>ression shock) 1n nonadla­be.tic fri0tion flow may be hendloo by the grarh:Lcal technique. Figure 8 ruay be used for nonadiacatlc flov! when the compression shock does not occur over toa long an interval.

Langley Memorial Aeronautical Laboratory NCttional Ad'li sor:;- 'Col1llYli ttee for AeronautIc:].

Langley Field, Va., jf.arch 16, 1.9L,h

-.

NAOA ARR No. 14016

1. Lelchuk.t V. L.: Beat. Transfer and HJ~draulic Flow Hesistance for Streama of High Velocity. HACA 'rM :Eo. 1054, 1943.

2. Fr6sse1, W.: Flow in Smooth Straight Pipes at Veloc~ties above and below Sound Velocity. NACA TM No. 81.Jl~1 1938.

23

3. Keenan, Joseph H., and Neumann, l:rn3st I?: ~leasure .. ments of Friction C08fficients 1n a Pipe for Subsonic ancl Supersonic Flow of Air. NACA ARR No. 3G13, 19h3.

,. ,. 4. Elias, Franz: The Transference of Heat from a Hot

Plate to an 1\:1 I' stream. NACL TM No. 614, 1931.

5. ~,r,cAdams, wtlliam H.: Heat 'I'ral1smission. Second ed., McGraw-Hi 11 Boo\{ 00., Inc. ,1942, p. 171.

6. F'luld Wot:ton Panel of the Aeronautical Reses.rcll Committee and Others: Modern Developments in Fluid Dynamics. Vol. II. S. Goldstein, ed., Clarendon Press (Oxford), 1938, --p. 612.

7. Busemann, 11..: G08d-yna;nik. Eandbuch der Experimenta1physl k, Ed. TV, 1. Tel1, Akael. Ver1agsgese11schaft m.b.li. (Leipzig), 1931, p. 366. _

8. Jung, Ingvar: Warmei).berg9.ng und TIei bungswlderatand bel Gasstrom'..mg in [oh:::8n bei Lohen Geschwlndig­kelten. Forscbungst...eft :CO. Beilage zu "Forschung a. d. Geb. d. Ingenieurwesens,:" Ausg. B, Dd. 7, Sept. -Oct. 1936, pp. 1-26.

, ' "

Fig. ,1

,/.035_'----.:.--r-...!..· --r-------r----r--~....., \ ,

,! '

!

1.0301--_._-+-":"--_' __ -~ __ l = O.57_-I-__ ~

/'025~-_4-___ /+--__ -+-~_~"':"-_--i

, 1.0Z0·~~_-+-__ -+ __ = 'K~~ €=-!

\ <, ' •

. /.0/5 ~ __ -I-___ I--__ -+-___ I--__ -I

,! ~~,,' '!

, 'I l ';/.42; I ,L _~

) lOIO I--__ ~-_-+__---.;-___+_--_I_--~

, '

I /

1.005~~" ____ ~~ __ ~ __ ~~~-----r----~

/.ooo'~ __ ~-+ ______ ~ ____ ~ ______ ~ ____ ~ , U j , L...... ~

Average squared flach number) 1'12 • " "" I' ,

, '". , 1 Figure 1.- Values of K from ~quation (1,3) for --r;:-power_

, veloci~y and stagnation-tempe~ature profiles.

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Mean-square 'Moen nu;n6e~(M:l)Vi.. ::; /.,

(al Subsonic heating.

Fi~ure 3.~ Isocline iolution o~ equation (15) for air in a tube at uniform ,wall' temperature.

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'NACA ARR No. L4C16

2.0

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Figure 4.- Von Karman-Nikuradse relation~hip between Reynqlda number and' skiri-frictio~ coefficient for flow IV sm~o~h tubes. '

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Fig. 5

1.0 :t.: I il I II II / I I -

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,/4 - . .14 -)iB: jvleon- square /vIQch number.; (M~

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,F, igure 5.-- Curves for constan, t,' , ,'y - 1.'M2 " " / , -', ·1 + 2 "

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to' be 'used wi th t'igure :3 for pressu:r;e-dr9P .calculations. _. ,

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- NACA ARR No. 'L4C16" . , ,

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16 ~.65r '\ 'Y \ /1\ \'l \ ~ II \ \ ' '\ \ \ 1\ \ ' \

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(alLow ~ange' of:~tibsonic entianceM~ah -nu~bers. , ~' '_ I J - ,.' _ _ ..... ~

.' . Figure 6.-- Plot of .equation (18) for adiabat.ic flow

- i:n smooth tubes. / , ," .

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Figure 6.~ Concluded. 'f

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Figure 7.- C6mparison:~f equa£ion (18) with experimental,data ftom-reference 3~ ,

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